This is only true if the ticket was used by the child when they were under 10. Sounds like that's not the case for the OP as they plan to save the unused child ticket from the 2nd week and the child will be over 10 when they use it.
I don't know the answer to the OP's question. But my guess based on what I've read is that an unused ticket would be credited at the price paid while a used ticket would be credited towards an upgraded ticket at the gate price. The child that's over 10 would not be able to use the 1-day child ticket so if you wait until later, I think the only option would be to get the price paid for it. If you want to avoid a price increase (assuming one doesn't happen before your trip), then I suggest upgrading the child 1-day to an adult 1-day while you're there. You'll only receive the price paid for the child ticket toward the purchase of and adult 1-day ticket, but if there's not a price increase by then, then the gate price for the child ticket would be the same as what you paid. Make sense?
Of course, you could always get a CM that doesn't strictly follow the rules and allows you to do something different, but I'm pretty sure what I've written would be Disney's guidelines/rules.